Grains of Wheat

Date: 07/14/99 at 00:21:41
From: Mostafa Khalafalla
Subject: Problem-solving
This is a question that I have already answered, but with a lot of
difficulty.
The person who invented the game of chess was said to have been
offered any payment he wanted. It is said that he asked for 1 grain of
wheat for the first square, 2 for the next, 4 for the third, and 8 for
the fourth, etc. for the 64 squares on the chessboard. How much wheat
did he receive?
First I tried doing it by hand. Then I searched all my maths books and
found nothing. Then I asked around the next day at school and found
that one of my friends had answered the question. I asked him how he
had done it and he went into a full-scale explanation of how he solved
it. He had done it on Excel using the formula for multiplication.
Could you please tell me an easier way to do it?

Date: 07/14/99 at 11:29:37
From: Doctor Rick
Subject: Re: Problem-solving
Hi, Mostafa.
I'm glad you want to know easier ways to do it by hand. Computers
make it easier to do a lot of arithmetic, but you don't get to know
numbers very well if you let the computer do all the work. Computers
can do it the hard way and still do it fast; if you do it by hand you
need to look for shortcuts, and that means getting to know numbers
and their patterns.
There is a pattern that can help you.
Square Grains Total so far
------ ------ ------------
1 1 1
2 2 +---3
|
3 4--+ 7
4 8 +--15
|
5 16--+ 31
Notice that the total so far is always 1 less than the number of
grains that will be put in the next square. This pattern is true for
any number of squares. You might want to investigate this to see why
it is true. If you have learned about binary numbers, you could write
the numbers in the table in binary, and remember that, for instance,
1111 (in binary) is 1 less than 10000. If this doesn't make sense to
you, don't worry about it, but look for other ways to think about the
pattern.
What this means is that you don't need to add up 64 numbers; you just
have to find the 65th number (how many grains would be put on the
65th square, if there were one). Subtract 1 from this number, and you
have the total number of grains on the first 64 squares.
The number of grains on square 65 is 2 * 2 * 2 ... * 2, with 64 2's.
(I use "*" for the multiplication sign.) Have you learned about
powers? We call this number 2 to the 64th power, written
64
2
With a keyboard, it's easier to write it as 2^64.
Powers follow patterns or rules, too. You can use one of these rules
to make it easier to calculate 2^64 - you don't have to do 63
multiplications. Here is the rule: if you square a power of 2, you
get another power of 2, with twice the power. For instance,
(2^4)^2 = 2^(4*2) = 2^8
If you haven't seen this, I'll show you why it works.
2^4 = 2*2*2*2 (2^4)^2 = (2*2*2*2) * (2*2*2*2)
\_____/ \___________________/
4 2's 8 2's
(2^4)^2 is a product of 8 2's, so it is 2^8.
Now watch how we can use this rule to calculate 2^64 quickly:
2^2 = 4
2^4 = (2^2)^2 = 4^2 = 16
2^8 = (2^4)^2 = 16^2 = 256
2^16 = ...
2^32 = ...
2^64 = ...
Finish those 3 lines, and you'll have the answer! The multiplications
get a lot bigger, but if you know how to estimate the size of a
product, you don't need to do all the work to get a good idea of how
big the total is.
I've gone over a lot of ideas quickly. I don't know how many of these
ideas you may already be familiar with. If you want to know more
about any of them, feel free to write back!
- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/